Epistasis and the Dynamics of Reversion in Molecular Evolution

Abstract

Recent studies of protein evolution contend that the longer an amino acid substitution is present at a site, the less likely it is to revert to the amino acid previously occupying that site. Here we study this phenomenon of decreasing reversion rates rigorously and in a much more general context. We show that, under weak mutation and for arbitrary fitness landscapes, reversion rates decrease with time for any site that is involved in at least one epistatic interaction. Specifically, we prove that, at stationarity, the hazard function of the distribution of waiting times until reversion is strictly decreasing for any such site. Thus, in the presence of epistasis, the longer a particular character has been absent from a site, the less likely the site will revert to its prior state. We also explore several examples of this general result, which share a common pattern whereby the probability of having reverted increases rapidly at short times to some substantial value before becoming almost flat after a few substitutions at other sites. This pattern indicates a characteristic tendency for reversion to occur either almost immediately after the initial substitution or only after a very long time.KEYWORDS weak mutation; fitness landscape; entrenchment; reversible Markov chain I N the context of evolutionary theory, reversion describes a population that returns to an ancestral character state (Porter and Crandall 2003). While many early (Dollo 1893;Muller 1939;Simpson 1953;Gould 1970) and more recent (Teotónio and Rose 2001;Collin and Miglietta 2008;Bridgham et al. 2009;Tan et al. 2011) discussions of reversion consider an environmental change that confers a selective advantage to an ancestral phenotype, reversion may also occur at the level of nucleic acids or protein sequences, with evolution proceeding under long-term purifying selection (Kimura 1983). Such reversions occur both because of the strictly limited number of character states (four possible nucleotides or 20 possible amino acids, Jukes and Cantor 1969) and because selection on molecular function may constrain a given position to only a subset of these possible character states (Rokas and Carroll 2008;Breen et al. 2012).It has long been hypothesized that epistatic interactions should lower the rate of reversion, rendering evolution effectively irreversible (Muller 1918(Muller , 1939. This issue has been especially important recently, due to ongoing debate in the field of protein evolution about how position-specific preferences for amino acids may change over time (Naumenko et al. 2012;Pollock et al. 2012;Ashenberg et al. 2013;Pollock and Goldstein 2014;Bazykin 2015;Doud et al. 2015;Goldstein et al. 2015;Risso et al. 2015;Shah et al. 2015;Usmanova et al. 2015). Specifically, several groups have suggested that once an amino acid substitution occurs at a particular position, epistatic interactions with subsequent substitutions at other positions should tend to increase the selective preference for the derived amino acid relative to the ancestral stat...

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